FOM: Finitely axiomatizable fragments of set theory
JoeShipman at aol.com
Tue May 15 13:43:16 EDT 2001
ZC and ZFC are not finitely axiomatizable. I know no consistent extension of ZF is finitely axiomatizable, but is it also true that no consistent extension of Z or ZC is finitely axiomatizable?
You can certainly replace Comprehension with "everything ZC proves about sets of rank less than omega+omega is true", after using finitely many instances of Replacement to define truth for V_(omega+omega), but this won't extend ZC because you won't get ZC-theorems about sets of higher rank. (On the other hand, you will have gotten a finitely axiomatizable theory that will suffice for 99+% of ordinary mathematics.) If you try instead to add an axiom of the form "there are no sets of rank omega+omega" you won't be able to define truth for V_(omega+omega).
There are also the finitely axiomatizable subtheories ZF1, ZF2, ...where ZFn contains the axiom "Every Pi_n theorem of ZFC is true" and the finitely many instances of Replacement needed to define truth for Pi_n formulas. Do these theories have any mathematical or metamathematical significance, or any nicer finite axiomatizations?
A final question: ZF proves the consistency of any finitely axiomatizable subtheory of itself. Do Z and PA also have this property?
-- Joe Shipman
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